Well, here's the answer (I think) to my previous question:

Which is larger, the set of all positive real integers or the set of all primes?

First we have to show that the set of all primes is, in fact, infinite. Fortunately, Euclid has done this for us:

http://primes.utm.edu/notes/proofs/infinite/euclids.htmlSo now I am going to claim they are equal. To do so, lets define a few conditions:

Let P be the set of all primes {p1, p2, ...p

m}, where p

i = p

j if and only if i = j.

Let S be the set of all positive real integers {1, 2, ...n}.

Now we will define R to be a transformation from S to P (R: S -> P) such that for some element

**s** in S, R(s) = p

sIt is easy to see that the mapping R: S -> P is one-to-one.

Consider

**a** and

**b**, elements of the set S. Then if R(

**a**) = R(

**b**), it implies

**a** =

**b**.

R(

**a**) = p

**a**R(

**b**) = p

**b**Then p

**a** = p

**b**, but p

i = p

j if and only if i = j, so we conclude that

**a** =

**b**, and R: S -> P is one-to-one.

It is also easy to see that R: S -> P is onto, for any p

k in P can be obtained by R(k).

Since R is one-to-one and onto, we can conclude that the sets are the same size.

QED